From daf02d38456fbb261c8b26f21834fe6249e5689b Mon Sep 17 00:00:00 2001 From: Vladyslav Shtabovenko Date: Fri, 29 Mar 2024 13:57:19 +0100 Subject: [PATCH] Section on renormalization updated. --- pages/Renormalization.tex | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/pages/Renormalization.tex b/pages/Renormalization.tex index 96edae5..dccc076 100644 --- a/pages/Renormalization.tex +++ b/pages/Renormalization.tex @@ -31,10 +31,10 @@ \subsection{Renormalization constants and mass generic masses, coupling constants and fields we can write \begin{align*} -g_{\textrm{bare}} &= \mu^{n \varepsilon} Z_g g_{\textrm{ren}} \\ -m_{\textrm{bare}} &= Z_m m_{\textrm{ren}} \\ -\psi_{\textrm{bare}} &= \sqrt{Z_m} \psi_{\textrm{ren}} \\ -A^\mu_{\textrm{bare}} &= \sqrt{Z_A} A^\mu_{\textrm{ren}} \\ +g_{\textrm{bare}} &= \mu^{n \varepsilon} Z_g g_{\textrm{ren}}, \\ +m_{\textrm{bare}} &= Z_m m_{\textrm{ren}}, \\ +\psi_{\textrm{bare}} &= \sqrt{Z_m} \psi_{\textrm{ren}}, \\ +A^\mu_{\textrm{bare}} &= \sqrt{Z_A} A^\mu_{\textrm{ren}}. \\ \end{align*} The renormalization scale \(\mu\) is needed to account for the fact, @@ -93,8 +93,8 @@ \subsection{Renormalization constants and mass where \(\delta Z_x\) contains poles in \(\varepsilon\) and possibly also finite pieces (depending on the chosen renormalization scheme). -Parametrically, \(Z_x\) is of order of the small coupling constant so -that we can ``expand'' in it as if \(Z_x \ll 1\). +Parametrically, \(\delta Z_x\) is of order of the small coupling +constant so that we can ``expand'' in it as if \(\delta Z_x \ll 1\). \hypertarget{examples-of-renormalized-and-counter-term-lagrangians}{% \subsection{Examples of renormalized and counter-term @@ -175,8 +175,8 @@ \subsection{Feynman rules}\label{feynman-rules}} Although those can be always derived by hand, doing so automatically is more convenient and allows to avoid many stupid mistakes. To this aim it -useful to employ \href{https://feynrules.irmp.ucl.ac.be/}{FeynRules} for -generating the corresponding FeynArts model. When writing down the +is useful to employ \href{https://feynrules.irmp.ucl.ac.be/}{FeynRules} +for generating the corresponding FeynArts model. When writing down the Lagrangian of our model we need to multiply every term in the counter term Lagragnian by \texttt{FR\$CT}, for example